The CRWN algorithm is described in Section 4.2.2. The main step is the choice of the regularization and its proximal map.
7.1.1
Problem specific regularization
As for the regularization, we consider two distinct options.
1) Our first option is the use of a total-variation (TV) regularization term to enhance the edges in the reconstructed image. Therefore, we set
Ψ2(c) =kLck1,1 (7.1)
withkLck1,1= ∑ik{Lc}ik1, where the sum is computed on all B-spline coefficients and{Lc}i∈ R2 is
the gradient vector of the image at position i. The discrete gradient operator L : RN → RN×2is computed
1A part of this chapter has been presented in [67]
according to Proposition 2 in [36]. Here, the regularization operator is the discrete gradient operator and the mixed`1− `1norm is chosen as the potential function. As the dual norm of the`1norm is`∞, the dual ball
is defined as
B∞,∞={p = [pT1, pT2, ..., pTN]T ∈ RN×2:
kpik∞≤ 1, ∀i = 1,2,...,N}. (7.2)
Therefore, the orthogonal projection of y∈ RN×2= [yT
1, yT2, ..., yTN]T onto this ball isey = PB∞,∞(y) with
[eyi]j= sgn([yi]j)min(|[yi]j|,1),
∀i = 1,2,...,N, j = 1,2, (7.3)
where[·]jis the jth entry of the corresponding vector andey = [eyT1,eyT2, ...,eyTN]T. This regularization is well-
matched to piecewise-constant images.
2) Owing to the fact that biological and medical specimens consist mostly of filament-like and compli- cated structures, we investigate higher-order extensions of total variation. We apply the Hessian Schatten- norm regularization (HS) as our second option. It can eliminate the staircase effect of TV regularization and results in piecewise-smooth variations of intensity in the reconstructed image. We set
Ψ2(c) =kHck1,S1, (7.4)
where H : RN → RN×2×2is the discrete Hessian operator andkHck
1,S1 is the mix of`1and nuclear norm.
The norm can be computed withkHck1,S
1 = ∑i(σ1,i+ σ2,i), where σ1,iand σ2,iare the singular values of
the Hessian matrix at position i. Therefore, the corresponding unit-norm dual ball is defined as B∞,S∞={p = [p
T
1, pT2, ..., pTN]T∈ RN×2×2:
kpikS∞ ≤ 1, ∀i = 1,2,...,N}, (7.5)
wherek·kS
∞ is the`∞-norm of the singular values of the corresponding matrix (for more details, we refer
the reader to [119]).
7.1.2
Parameter setting
The proposed algorithm has several parameters.
• Parameters λ1and λ2: We use the approach proposed in [36]; λ1= 10−5 and λ2= 10−4kgk. The
experimental results suggest that this choice of parameters yields the optimal performance.
• Parameter µ: This parameter affects the convergence speed of ADMM. Since the algorithm is not too sensitive to it, we use a fixed value (µ= 1).
• Parameter λ : This is a parameter of the proximal map operator in (4.32). Since the second step of
ADMM is solving (4.18), we have λ= λ2/µ.
• Lipschitz constant L: The Lipschitz constant of ∇ f(p) =−λ RPC(z− λ RTp) is approximated by
the Lipschitz constant of the same operator without the convex projectionPC since the projection on the convex set is firmly non-expansive. Thus,
where λmax(A) is the maximum eigenvalue of the matrix A. For our regularization scheme λmax(RRT)≤
γ where γ= 8 for the TV regularization for two-dimensional problems, and its value is 64 for the HS regularization as computed in [119].
• Parameter τ: We set it to τ= 1/10× L−1.
7.1.3
Experimental result
We compared the proposed algorithm to FBP and to ADMM-PCG, which appears to be the current state of the art for the reconstruction of X-ray-DPCI tomograms [36].
(a)
(b)
Figure 7.1: Two reference samples (a) and (b).
All experiments involved real data acquired at the TOMCAT beam line of the Swiss Light Source at the Paul Scherrer Institut in Villigen, Switzerland. The common approach for these experiments is to use a reconstruction from a large number of projections as a reference for evaluating results obtained with signifi- cantly fewer projections. In addition, the convex constraints that we apply are the positivity of the refractive index combined with the support-related constraint that the solution should be zero outside the tube that contains the specimen.
In order to identify the benefits of the proposed algorithm (CRWN), we first tested the algorithms under extreme conditions: We used only 72 projections as input, while the reference was reconstructed from 1,200 projections. For this first experiment we used a phantom that was composed of a tube and three cylinders containing liquids with different refractive indices as shown in Figure 7.1(a).
The performance of different algorithms are compared in Table 7.1. Clearly, the new method outper- forms ADMM-PCG [36]. Applying the convex constraint improves the signal-to-noise ratio (SNR) and the structural similarity index measure (SSIM) [116] even further. The result of the algorithm proposed in [120] is the same as CRWN-TV without CC, but it is slower since it uses FISTA. As expected, owing to the piecewise-constant structure of the sample, TV outperforms HS regularization.
The proposed algorithm (CRWN) Constrained Unconstrained TV HS TV [120] HS ADMM-PCG [36] FBP Phantom SNR(dB) 27.49 23.91 25.89 21.82 17.62 2.177 SSIM 0.509 0.369 0.339 0.196 0.145 0.07 Scaffold SNR(dB) 25.34 25.58 22.91 22.25 20.09 6.45 SSIM 0.673 0.699 0.574 0.566 0.512 0.186 Scaffold ROI SNR(dB) 26.51 27.05 23.78 23.75 23.58 23.09 SSIM 0.968 0.974 0.944 0.958 0.852 0.516
Table 7.1: Performance of different reconstruction techniques that have been applied on Phantom and Scaffold samples.
We conducted another experiment with a coronal section of a scaffold that is used for surgery. The reference image was built from 2,000 projections as depicted in Figure 7.1(b). The algorithms were then benchmarked on a subset of 250 projections. Although these conditions are less severe, FBP still produces high-frequency patterns that are visible in Figure 7.2(a). ADMM-PCG almost completely suppresses these artifacts, at the expense of light smoothing as shown in Fig. 7.2(b). Overall, CRWN yields sharper images, as shown in Figure 7.2(c) and 7.2(d), which is also reflected by the quality metrics. In addition, Hessian type regularization eliminates the staircase effect of TV which is more visible in the selected region of interest.
It is seen in Figure 7.3(a) that CRWN is significantly faster at minimizing the cost functional than the standard FISTA algorithm. In addition, it appears that the convergence speed is not very sensitive to the number of inner iterations as we use a warm initialization. We illustrate in Figure 7.3(b) the robustness of CRWN with respect to the number of projections in terms of SNR. Owing to the poor performance of FBP in reconstructing boundaries, we compute the SNR for the region specified by a dashed circle in Figure 7.1(b).